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[Reprinted  from  The  American  Mataematical  Monthly,  Vol.  XXXV,'  Apri!,M9l?  |'\  J  *?  ' 


NOTE  ON  SOME  APPLICATIONS  OF  A  GEOMETRICAL  TRANS- 
FORMATION TO  CERTAIN  SYSTEMS  OF  SPHERES.^ 

By  dr.  henry  W.  STAGER,  Fresno,  California. 

Ill  a  paper  in  Volume  VI  of  the  Proceedings  of  the  Edinburgh  Mathematical 
Society,  Professor  Allardice  considers  the  transformation  in  piano: 

Let  I  be  a  fixed  straight  line,  C  any  plane  curve,  t  a  tangent  to  C  meeting  / 
in  X  and  making  an  angle  6  with  /;  then  C,  the  transformed  curve  of  C,  is  the 
envelope  of  a  straight  line  t'  through  X  making  an  angle  0  with  I,  where  6  and  0 
are  connected  by  the  relation 

tan  |0  =  k  tan  ^6. 

The  length  of  the  tangent  from  any  point  in  I  to  C  remains  unaltered  by  the  trans- 
formation. A  similar  method  is  applicable  in  space  and  the  following  paper 
gives  some  of  the  results  in  its  applications  to  certain  systems  of  spheres.  The 
method  of  transformation  is  as  follows: 

Let  a  be  a  fixed  plane,  S  any  surface,  and  ^  a  plane  tangent  to  S  intersecting 
a  in  the  line  i  and  making  with  a  an  angle  6.  If  (3'  be  a  plane  through  i  making 
with  a  an  angle  0,  determined  by  the  relation 

(1)  tan  ^(f)  =  k  tan  hd, 

then  the  envelope  of  j8'  will  be  defined  as  the  transformed  surface  of  S,  or,  more 
simply,  the  "transform  of  S."  ^.. 

From  the  given  relation,  tan  ^0  —  k  tan  |0,  and  the  identity, 

_     2  tan  (/)/2 

we  find  that 

2k  tan  d 

(2)  tan  (j) 


±  (1  -  F)  Vl  +  tan2  d  +  1  +  F ' 

It  is  evident  from  the  nature  of  this  relation  that  for  every  value  of  k  there  are 
two  values  of  tan  0,  according  as  we  consider  the  angle  afi  as  d  or  (6  -{-  ir). 

For  simplicity,  we  will  use  Cartesian  coordinates  to  find  the  equation  of  the 
I  transform  of  a  given  sphere  from  its  equation,  and  we  will  take  the  .r^-plane  as 
the  plane  of  transformation,  a.  Let  6  be  the  angle  formed  by  the  .r?/-plane  and 
the  plane  ux  +  vy  +  ivz  —  1  =  0,  and  0  be  the  angle  formed  by  the  a-z/-plane 
and  the  plane  ux  +  vy  +  2^12  —  1  =  0.  (Since  the  intercepts  of  both  planes  on 
the  .T-axis  and  on  the  y-ax\s  are  equal,  respectively  l/u  and  l/v,  they  intersect 
the  .T2/-plane  in  the  same  line,  as  required  by  the  transformation.) 

'  Presented  to  the  San  Francisco  Section  of  the  American  Mathematical  Society,  April 
12,  1913. 


'  1*55  •'  APPUCATIONS   OF  A   GEOMETRICAL  TRANSFORMATION. 

From  the  formula  for  the  angle  between  two  planes, 

0    _  (J?(7^  -  B'Cf  +  {CA'  -  CAY  +  {AB'  -  A'Bf 
tan-     -  ^^j,  _^  ^^,  _^  ^^,^2 

we  have 

*?/       ■  I  ■    ill"  97       I   I        -i*** 

tan-  0  = ^ —  and  tan^  (b  = , — . 

Substituting  these  values  in  equation  (2)  and  solving  for  w,  we  obtain 

(1  +  /r)wi  ±  (1  -  F)  a/m2  _^  ^^+^2 
(3)  ^.  = 2^ , 

or, 


,^^           ,          2(1  +  A;4)Wi2  +    (1    -  F)2(W2  +  ^2)  _i.  2^,(1    _   Jt4)  Vw2  +  ^2  _|.  ^^2 
(4)       ^2  = _^^ , 

Now,  let  the  center  of  the  sphere  lie  in  the  2-axis  and  its  equation  will  be 

(5)  x^+y''+z''  +  2nz+  d=  0. 

Expressing  the  condition  that  the  plane  ux  -\-  vy  -\-  wz  —  1  =  0  he  tangent 
to  this  sphere,  we  have 

(6)  (d  -  71^1^  +  (d  -  n^y  +  dw'^  +  2nw  +1  =  0, 

which  is  the  tangential  equation  of  the  sphere. 

Substituting  the  values  of  w  and  w-  found  in  (3)  and  (4)  and  then  simplifying 
and  factoring  the  result,  we  have  the  equation  of  the  transform  of  the  given  sphere 

[{(F  +  lyd  -  2n^{¥  +  1)  +  2n{h^  -  1)  -in^  -  d]{iv'  +  v")  +  Ulc^w^- 

+  4F  +  4/j{n(P  +  1)  +  (F-  -  1)  4n^  -  d]wA  X  [{(/r  +  l)-(i 

(7)  , 

-  2n\k'  +  1)  -  2/t(/.-'  -  1)  V/i2  -  d^iv"  +  r)  +  Ukhci'^  +  4^^ 

+  4/j{n(F  +  1)  _  (/,2  _  1)  V^i^  _  d]ic?[  =  0. 

This  equation  breaks  up  into  two  equations  which  are  the  tangential  equations 
of  two  spheres,  and  may  be  written  in  the  form 

(8)  yli(w2  ^  ^2)  ^  Q^^,2  _^  2NiWi  +  D  =  0; 

(9)  ^2(^2  +  v"")  +  Cwi"  +  2A^2Wi  +  D=  0. 
The  Cartesian  equations  of  these  forms  are 

(10)  {CD  -  Ni')x^  +  (CD  -  iVi2)7/2  +  AiDz""  +  .4iC  +  2yli.Yi2  =  0; 

(11)  {CD  -  N2V  +  {CD  -  N.})y''  +  AnDz-  +  J-.C  +  2/I2.V02  =  0, 
where,  from  the  actual  values  of  the  coefficients  involved. 


TO   CERTAIN   SYSTEMS   OF  SPHERES.  156 

(CD  -  Ni")  =  AiD,        and        {CD  -  Ni")  =  A^D. 
Simplifying  and  substituting,  we  have  finally 


(12)  4F(.t2  +  2/2  +  S-)  +  ik{n(F"  +  1)  +  (/v-  -  1)  Vw^  -  d}z  +  4<fF  =  0; 

(13)  4F(a:2  +  r  +  S")  +  ^k{n{k''  +  1)  -  (F  -  1)  ^^-^}z  +  4dA;2  =  Q: 

the  equations  of  two  spheres  whose  centers  lie  in  the  z-axis.  Hence,  in  general, 
a  sphere  is  transformed  into  two  spheres  whose  centers  lie  in  a  line  perpendicular 
to  the  plane  of  transformation  and  passing  through  the  center  of  the  given  sphere. 
Now,  let  p  represent  the  radius  of  the  given  sphere  and  5  the  distance  of  its 
center  from  the  plane  of  transformation;  also,  let  r'  and  r"  and  d'  and  d"  repre- 
sent respectively  the  radii  and  distances  from  the  plane  of  transformation  of  the 
centers  of  the  transformed  spheres.  The  equations  of  the  latter  may  then  be 
written 

(14)  x'  +  2/2  _^  .2  _^  [_  5(^.  +  1//,)  ^  (/,  _  i|^.)^],  +  d=0; 

(15)  x^  +  r  +  2'  +  [-  8{k  +  l/k)  -  {k  -  \lk)p\z  +  (i  =  0: 
whence  the  resulting  relations: 

(^'  4-  /  =  (5  -  p)k;  d"  +  t"  =  (5  +  p)k; 

(16) 

d'  -  r'  =  (5  +  p)lk;        d"  -  r"  =  (5  -  p)lk. 

We  will  next  consider  for  what  values  of  k  the  transformed  spheres  will 
degenerate  into  points.     It  is  evident  that  we  must  consider  each  case  separately. 
In  the  first  case  we  have 


r'  = 


-^{(5+p)  -kH8-  p)}\=0, 


whence 

(17)  'F-  =  (5  +  p)/(5  -  p). 

Here  we  have  two  real  values  for  k,  provided  the  given  sphere  does  not  cut  the 
plane  of  transformation;  and  a  zero  or  infinite  value  for  k  if  the  given  sphere  is 
tangent  to  that  plane.     The  corresponding  values  of  the  distance  are 


d'  =  T  V52  -  p 


2 


or  in  terms  of  the  original  coefficients, 

(18)  d'  =  q=  V^. 

In  the  second  case,  by  similar  methods,  we  find  that 

(19)  P  =  (5  -  p)/(5  +  p), 
and 

(20)  d"  =  =F  V52  -  p\        or,        d"  =  T  V^. 


157  APPLICATIONS    OF    A   GEOMETRICAL  TRANSFORMATION, 

It  follows  that,  in  general,  a  sphere  may  be  transformed  into  a  point  for  any 
one  of  four  values  of  k,  the  points  being  the  same  in  pairs,  and  all  four  being  equi- 
distant from  the  plane  of  transformation,,  ope  pair  on  each  side.  It  is  to  be 
especially  noted  that  while  each  of  the  two  transformed  spheres  may  degenerate 
into  points,  both  spheres  cannot  become  points  at  the  same  time,  unless  either 
p  =  0,  or  5  =  0;  i.  e.,  when  the  given  sphere  is  itself  a  point  and  this  point  may 
be  transformed  into  its  image  with  respect  to  the  plane  of  transformation,  or 
when  the  center  of  the  given  sphere  lies  in  the  plane  of  transformation  and  the 
points  are  imaginary.  We  may  also  note  that  a  point  can  be  transformed  into 
only  one  sphere. 

We  have  already  seen,  equations  (12)  and  (13),  that  for  the  same  value  of  h 
a  given  sphere  may  be  transformed  into  two  distinct  spheres.  These  two  spheres 
evidently  result  from  the  two  values  of  0,  according  as  we  consider  the  angle 
a/3  as  d  or  {B-\-  -k).  In  case  the  angle  is  B  we  will  say  that  the  transformed 
sphere  is  traced  out  by  a  direct  movement,  and  where  the  angle  is  {Q  +  tt)  we 
will  say  that  it  is  traced  out  by  an  indirect,  or  inverse,  movement.  It  is  evident 
that  if  we  transform  two  spheres,  hoth  directly,  or  both  inversely,  the  direct 
common  tangent  planes  to  the  two  spheres  become  common  tangent  planes  to 
the  transformed  spheres,  while  if  one  sphere  is  transformed  directly  and  the 
other  inversely,  the  transverse  common  tangent  planes  become  common  tangent 
planes  to  the  transformed  spheres. 

Forming  the  equation  of  the  radical  plane  of  the  given  sphere  and  either  of 
its  transforms,  we  have 

z  =  0, 

which  is  also  the  plane  of  transformation.  It  follows  that  the  plane  of  transforma- 
tion is  the  radical  plane  of  the  given  sphere  and  its  transforms,  and  that  the 
length  of  a  tangent  line  from  any  point  in  the  plane  of  transformation  to  the 
sphere  is  unaltered  by  the  transformation.  Hence  the  distance  between  the 
points  of  contact  of  common  tangent  planes  to  two  spheres  remains  unaltered. 
This  is  the  important  property  of  the  transformation.  By  transforming  several 
spheres  into  points  at  the  same  time,  relations  and  properties  of  spheres  may  be 
obtained  directly  from  known  relations  between  points. 

For  two  or  more  spheres  to  be  transformed  simultaneously  into  points,  it  is 
evidently  sufficient  that  the  values  of  Ic-  be  equal.  Considering  the  case  of  two 
spheres  with  radii  p'  and  p"  and  with  the  distances  of  their  centers  from  the 
plane  of  transformation,  5'  and  5",  respectiveh',  we  have 

p  =  (5'  -  p')/(5'  +  p')     and     P  =  (d"  -  p")l{h"  +  p"); 
whence 

(5'  -  p')/(5'  +  p')  =  (5"  -  p")l{b"  +  p"),     or    575"  =  p'lp". 

This  is  the  condition  that  the  plane  of  transformation  pass  through  the  direct 
center  of  similitude  of  the  two  spheres.  Using  the  reciprocal  values  for  k"^  we 
obtain  the  same  result.     Finally,  let 


TO  CERTAIN  SYSTEMS  OP  SPHERES.  158 

F  =  (5'  -  p')/(5'  +  p')     and     P  =  (5"  +  p")/(5"  -  p")\ 
whence 

575"  =  -  p'Ip": 

the  condition  that  the  plane  of  transformation  pass  througli  the  inverse  center 
of  simihtude  of  the  two  spheres.  We  conclude  that  two  spheres  may  be  trans- 
formed simultaneously  into  points  if  the  plane  of  transformation  contain  a 
center  of  similitude  of  the  two  spheres.  Likewise,  if  the  plane  of  transformation 
contain  any  one  of  the  four  axes  of  similitude  of  three  spheres,  these  spheres  will 
all  be  transformed  into  points  by  the  same  transformation.  Further,  if  the  plane 
of  transformation  be  any  one  of  the  eight  planes  of  similitude  of  four  spheres/ 
the  four  spheres  may  be  transformed  into  points  simultaneously;  and,  finally, 
any  number  of  spheres  which  have  a  common  plane  of  similitude  may  be  so 
transformed.  Two  spheres  on  the  same  side  of  the  plane  of  transformation  will 
be  transformed,  either  both  directly,  or  both  inversely;  while  of  two  on  opposite 
sides,  one  will  be  transformed  directly,  the  other  inversely. 

By  giving  k  every  possible  value  a  given  sphere  may  be  transformed  into  an 
infinite  series  of  spheres  which,  by  virtue  of  the  properties  of  the  plane  of  trans- 
formation as  the  radical  plane  of  a  sphere  and  its  transforms,  is  cut  orthogonally 
by  a  system  of  spheres.  Then,  since  the  given  sphere  is  one  of  the  series,  if  we 
transform  the  whole  system  for  the  same  value  of  h,  it  will  transform  into  itself. 

It  may  be  noted  at  this  point  that  the  results  here  obtained  by  analytic 
methods  may  readily  be  obtained  by  synthetic  methods,  and  in  the  sequel  they 
will  be  interpreted  in  accordance  with  the  particular  problem  under  consideration. 
In  all  problems  which  follow  it  w^ill  be  necessary  to  associate  only  those  common 
tangent  planes  and  distances  between  points  of  contact  of  common  tangent 
planes  to  two  spheres  which  determine  the  centers  of  similitude  in  the  planes 
of  similitude  used  as  planes  of  transformation.  Thus  we  will  associate  with  the 
plane  of  similitude  passing  through  the  six  direct  centers  of  similitude  of  four 
spheres  the  external  common  tangent  planes  of  each  pair  of  spheres,  and  only 
the  external  common  tangent  planes  of  these  spheres. 

Some  Applications  of  the  jNIethod. 

1.  The  condition  that  four  planes  passing  through  a  point  should  be  tangent 
to  the  same  sphere  is  given  by  the  equation 

tan  \  (j)'  tan  ^  </>"  =  tan  ^  d'  tan  ^  6", 

where  0',  cf)",  6',  6"  are  the  angles  formed  by  the  given  planes  and  the  plane 
determined  by  the  lines  of  intersection  of  the  two  pairs  of  opposite  planes. 

1  The  twelve  centers  of  similitude  of  four  spheres  lie  in  sets  of  six  in  a  plane:  the  six  direct 
centers  lie  in  a  plane;  the  three  direct  centers  of  any  three  spheres  lie  in  a  plane  with  the  three 
inverse  centers  not  paired  with  them;  and  any  two  direct  centers,  using  each  sphere  only  once, 
lie  in  a  plane  with  the  four  inverse  centers  not  paired  with  them.  Thus  there  are  eight  planes 
of  similitude,  one  containing  all  the  direct  centers;  four  containing  three  direct  and  three  inverse 
centers;  and  three  containing  two  direct  and  four  inverse  centers. 


159 


APPLICATIONS   OF  A   GEOMETRICAL   TRANSFORMATION. 


Transform  the  sphere  into  a  point  with  the  external  diagonal  plane  as  the 
plane  of  transformation,  and  let  the  angles  of  the  transformed  planes  with  this 
plane  of  transformation  be  0  and  6,  corresponding  respectively  to  the  angles 
4)' ,  (f)"  and  6',  6".     We  then  have  the  following  relations: 

tan  I  0  =  A-  tan  \  <^';  tan  \  d  =  k  tan  h  6'; 

tan  I  (0  +  tt)  =  /.•  tan  |  (j>";         tan  ^  (d  +  t)  =  k  tan  |  d". 

Eliminating  cj),  6,  h  frdin  these  equations,  we  have  the  given  condition. 

2.  If  five  spheres  have  a  common  plane  of  similitude  they  may  be  transformed 
into  five  points  by  the  same  transformation.  Therefore  the  distances  between 
the  points  of  contact  of  the  common  tangent  planes  of  each  pair  of  spheres  satisfy 
the  relation  connecting  the  ten  straight  lines  joining  five  points  in  space. 

Let  di,  do,  dz,  •  •  ■  f/io  be  the  distances.  Then  the  relation  is  given  by  the 
determinant^ 


0 

rfr 

d-z' 

^3^ 

d:~ 

di' 

0 

d/ 

d,' 

d^' 

d,' 

d,' 

0 

de 

d<f 

dz' 

d,' 

ds' 

0 

d,o' 

d^ 

d,' 

d,' 

d,^ 

0 

1 

1 

1 

1 

1 

0 

0. 


3.  If,  in  addition  to  the  conditions  of  No.  2,  the  five  spheres  also  touch  a 
sixth  sphere,  the  ten  distances  will  satisfy  the  additional  relation  connecting  five 
points  lying  on  the  surface  of  a  sphere. 

Transforming  the  five  spheres  into  points  as  above,  they  must  lie  on  the 
surface  of  the  transform  of  the  sixth  sphere  and  will  therefore  satisfy  the  addi- 
tional relation  given  by  the  determinant^ 


0 

d^ 

d-f 

d^ 

d^ 

d,' 

0 

d^ 

d,' 

d,' 

di" 

d,' 

0 

di 

d,-^ 

d,' 

d,' 

^8^ 

0 

did 

d,' 

d,' 

d,' 

c/io^ 

0 

=  0. 


4.  The  radius  11  of  a  given  sphere  S  tangent  to  four  given  spheres  may  be 
found  in  terms  of  R,  the  distance  5  of  the  center  of  iS  from  the  plane  containing 
the  six  direct  centers  of  similitude  of  the  four  spheres,  the  distances  between 
the  j)oints  of  contact  of  common  tangent  planes  to  pairs  of  spheres,  and  the 

'  Cayluy,  "CuUcclcd  Mathematical  Papers,"  Volume  I,  p.  1  et  seq. 


::• 


:\' 


TO   CERTAIN   SYSTEMS   OF   SPHERES.  160 

radius  and  distance  from  its  center  to  the  plane  of  similitude  of  any  one  of  the 
given  spheres. 

If  we  transform  the  four  spheres  into  points  by  the  same  transformation, 
these  four  points  will  lie  on  the  transform  of  S.     Hence 


^  _  1     / (2aa')[n(-  aa'  +  bh'  +  cc')] 

^'  ~  2  \l.Wa'\W  +  c2  -  a2  +  h'''  +  c'^  -  a'^)]  -  {cC-W  -  i:a'b"c") ' 

where  a,  a',  etc.,  are  the  distances  between  the  points  of  contact  of  the  common 
tangent  planes  of  two  pairs  of  spheres  with  no  sphere  in  common,  and  where  2 
and  TT  represent  cyclo-symmetrical  functions  only.     But 


R'  = 
where 


^   {5(P-1)T(F+1)/^} 


21c 
F  =  (^^  _  ri)l(di  +  n),        or        P  =  (di  +  ri)l(di  -  n), 


from  equations  (17),  and  (19). 
Hence 

R=\-  5(ri/(^i)  ±  2{(^dx'-r,')fd}W\. 

5.  The  locus  of  the  center  of  a  sphere  S  such  that  the  distances  between  the 
points  of  contact  of  the  common  tangent  planes  to  S  and  four  fixed  spheres  are 
equal  is  a  straight  line  perpendicular  to  a  plane  of  similitude  of  the  four  spheres. 

Transforming  the  fixed  spheres  into  points,  the  transformed  spheres  S'  will 
form  a  system  of  spheres  Avith  a  common  center;  namely,  the  center  of  the 
sphere  determined  by  the  four  points.  INIoreover,  all  the  spheres  S'  with  real 
tangents  to  the  four  points  lie  within  this  sphere;  consequently  the  distance 
between  the  points  of  contact  cannot  exceed  the  radius  of  this  sphere,  and  hence 
it  is  a  maximum  when  S'  becomes  a  point;  i.  e.,  when  the  four  fixed  spheres  and  S 
have  a  common  plane  of  similitude.  The  theorem  follows  directly  from  the  fact 
that  the  system  S'  is  a  system  of  concentric  spheres.  There  are  eight  cases  for  con- 
sideration corresponding  to  the  eight  planes  of  similitude  of  the  four  fixed  spheres. 

6.  The  centers  of  the  spheres  tangent  to  four  given  spheres  lie  in  pairs  on 
eight  straight  lines  passing  through  the  radical  center  and  perpendicular  to  one 
of  the  planes  of  similitude  of  the  four  spheres. 

This  is  only  a  special  case  of  the  preceding.  It  is  evident  that  the  radical 
plane  of  any  pair  of  the  tangent  spheres  is  the  plane  of  similitude,  which  is 
perpendicular  to  the  line  joining  the  centers  of  the  two  spheres.  The  results 
of  this  theorem  can  be  employed  in  the  construction  of  the  spheres  tangent  to 
four  given  spheres. 

7.  If  >S  is  a  sphere  tangent  to  four  given  spheres,  which  are  tangent  to  one 
another,  the  five  spheres  have  common  planes  of  similitude. 

Transforming  the  four  spheres  into  points,  the  points  will  coincide,  since  each 
sphere  touches  one  of  the  others.     But  the  four  points  must  lie  on  S',  the  trans- 


161  APPLICATIONS   OF  A   GEOMETKICAL  TRANSFORMATION. 

form  of  S,  and  therefore  S'  must  also  reduce  to  a  point  simultaneously  with  the 
other  four;  hence  the  result. 

8.  The  locus  of  the  centers  of  all  spheres  which  touch  a  given  sphere  and 
have  a  common  plane  of  similitude  with  it  is  an  ellipsoid  of  revolution. 

Let  r  be  the  radius  and  d  be  the  distance  from  the  common  plane  of  similitude 
of  the  center  of  the  given  sphere,  and  let  p  and  8  be  the  radius  and  distance 
respectively  of  the  variable  sphere.  Transform  the  spheres  into  points  for  the 
same  value  of  k  and  we  have 

7  (J 

d'  =  ^  (k  +  lA)  -lik-  l/k) •        6'  =  2  (/.  +  1/^)  -  2  ^^'  ~  ^'^'^' 

But,  since  all  the  spheres  are  tangent  to  the  given  sphere,  the  points  into  which 
they  transform  are  coincident,  and  therefore,  d'  =  5'.     Hence 

{k  -  HMr  +p)  =  (^d+b-  Y^^^  {k  +  m, 

or,  (r  +  p)/(5  +  c)  =  e,  where  e  and  c  are  constants. 

Therefore,  the  locus  of  the  centers  of  the  spheres  is  the  locus  of  a  point, 
the  ratio  of  whose  distances  from  the  center  of  the  given  sphere  to  its  distance 
from  a  plane  parallel  to  and  at  the  distance  c  from  the  common  plane  of  similitude 
is  constant;  i.  e.,  an  ellipsoid  of  revolution. 


Caylord  Bros. 

Makers 

Syracuse,  N.  Y. 

MI.  IMI.  21, 19« 


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